Here are examples corresponding to some of the topics listed in the course outline:

### 1. **Radian and Degree Measure**
- **Example**: Convert \( 120^\circ \) to radians.
  - **Solution**: Use the conversion formula \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180^\circ} \).
    \[
    120^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{3} \text{ radians}
    \]

### 2. **Verifying Trigonometric Identities**
- **Example**: Verify the identity \( \sin^2\theta + \cos^2\theta = 1 \).
  - **Solution**: This is a fundamental Pythagorean identity, which holds true for any angle \( \theta \). It's based on the relationship between the sides of a right triangle and the unit circle.

### 3. **DeMoivre's Theorem**
- **Example**: Use DeMoivre's Theorem to find \( (1 + i)^4 \), where \( i \) is the imaginary unit.
  - **Solution**:
    1. Express \( 1 + i \) in polar form: \( r = \sqrt{2} \), \( \theta = \frac{\pi}{4} \).
    2. Apply DeMoivre's Theorem: \( (r(\cos\theta + i\sin\theta))^4 = r^4(\cos(4\theta) + i\sin(4\theta)) \).
    3. Simplify: \( 2^2(\cos(\pi) + i\sin(\pi)) = 4(-1 + 0i) = -4 \).

### 4. **Solving Trigonometric Equations**
- **Example**: Solve \( 2\sin x - 1 = 0 \) for \( x \) in the interval \( [0, 2\pi] \).
  - **Solution**:
    1. Isolate \( \sin x \): \( \sin x = \frac{1}{2} \).
    2. Find the general solution: \( x = \frac{\pi}{6}, \frac{5\pi}{6} \) within the given interval.

### 5. **Trigonometric Functions and the Unit Circle**
- **Example**: Find the sine and cosine of \( 225^\circ \) using the unit circle.
  - **Solution**:
    1. \( 225^\circ \) corresponds to an angle in the third quadrant.
    2. Reference angle: \( 225^\circ - 180^\circ = 45^\circ \).
    3. Sine and cosine values: \( \sin 225^\circ = -\frac{\sqrt{2}}{2} \), \( \cos 225^\circ = -\frac{\sqrt{2}}{2} \).

Would you like a more detailed explanation of any of these examples?

### 5 Follow-Up Questions:
1. Can you find the radian measure for \( 330^\circ \)?
2. Would you like to verify another trigonometric identity, such as \( 1 + \tan^2\theta = \sec^2\theta \)?
3. How would you use DeMoivre's Theorem to find the cube roots of \( 8 \)?
4. Can you solve the equation \( \cos 2x = \frac{1}{2} \) within \( [0, 2\pi] \)?
5. What are the sine and cosine values for \( 135^\circ \) using the unit circle?

**Tip:** Practice converting between degrees and radians regularly to become more comfortable with both units of angle measurement.

Explore examples of trigonometry problems including converting degrees to radians, verifying trigonometric identities, and applying DeMoivre's Theorem. Understand key formulas and theorems with detailed solutions.

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